Spread assumed to be equal to mean. (\(\phi = 1\))
Dispersion
Over-dispersion
Sample more varied than expected from its mean
variability due to other unmeasured influences
quasi- model
due to more zeros than expected
zero-inflated model
Logistic regression
Example data
y x
1 0 1.024733
2 0 2.696719
3 0 3.626263
4 0 4.948643
5 0 6.024718
6 0 6.254113
Logistic regression
Fit model
>dat.glmL <-glm(y ~x, data = dat, family ="binomial")
Logistic regression
Explore residuals
>par(mfrow=c(2,2))
>plot(dat.glmL)
Logistic regression
Explore goodness of fit
Pearson’s \(\chi^2\) residuals
>dat.resid <-sum(resid(dat.glmL, type ="pearson")^2)
>1 -pchisq(dat.resid, dat.glmL$df.resid)
[1] 0.8571451
Deviance (\(G^2\))
>1-pchisq(dat.glmL$deviance, dat.glmL$df.resid)
[1] 0.8647024
Logistic regression
Explore model parameters
Slope parameter is on log odds-ratio scale
>summary(dat.glmL)
Call:
glm(formula = y ~ x, family = "binomial", data = dat)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.97157 -0.33665 -0.08191 0.30035 1.59628
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -6.9899 3.1599 -2.212 0.0270 *
x 0.6559 0.2936 2.234 0.0255 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 27.526 on 19 degrees of freedom
Residual deviance: 11.651 on 18 degrees of freedom
AIC: 15.651
Number of Fisher Scoring iterations: 6
Logistic regression
Quasi \(R^2\)\[quasi R^2 = 1-\left(\frac{deviance}{null~deviance}\right)\]
